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We study the computational complexity of (deterministic or randomized)algorithms based on point samples for approximating or integrating functionsthat can be well approximated by neural networks. Such algorithms (mostprominently stochastic gradient descent and its variants) are used extensivelyin the field of deep learning. One of the most important problems in this fieldconcerns the question of whether it is possible to realize theoreticallyprovable neural network approximation rates by such algorithms. We answer thisquestion in the negative by proving hardness results for the problems ofapproximation and integration on a novel class of neural network approximationspaces. In particular, our results confirm a conjectured and empiricallyobserved theory-to-practice gap in deep learning. We complement our hardnessresults by showing that approximation rates of a comparable order ofconvergence are (at least theoretically) achievable.

Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces